Stabilised explicit Adams-type methods
Abstract
In this work we present explicit Adams-type multi-step methods with extended stability intervals, which are analogous to the stabilised Chebyshev Runge – Kutta methods. It is proved that for any k ≥ 1 there exists an explicit k-step Adams-type method of order one with stability interval of length 2k. The first order methods have remarkably simple expressions for their coefficients and error constant. A damped modification of these methods is derived. In the general case, to construct a k-step method of order p it is necessary to solve a constrained optimisation problem in which the objective function and p constraints are second degree polynomials in k variables. We calculate higher-order methods up to order six numerically and perform some numerical experiments to confirm the accuracy and stability of the methods.
References
- Hairer E, Wanner G. Solving ordinary differential equations II: stiff and differential-algebraic problems. Berlin: Springer; 1996. 614 p. (Springer series in computational mathematics; volume 14). DOI: 10.1007/978-3-642-05221-7.
- Lebedev VI. How to solve stiff systems of differential equations by explicit methods. In: Marchuk GI, editor. Numerical methods and applications. Boca Raton: CRC Press; 1994. p. 45–80.
- Sommeijer BP, Shampine LF, Verwer JG. RKC: an explicit solver for parabolic PDEs. Journal of Computational and Applied Mathematics. 1998;88(2):315–326. DOI: 10.1016/S0377-0427(97)00219-7.
- Abdulle A, Medovikov AA. Second order Chebyshev methods based on orthogonal polynomials. Numerische Mathematik. 2001;90(1):1–18. DOI: 10.1007/s002110100292.
- Abdulle A. Fourth order Chebyshev methods with recurrence relation. SIAM Journal on Scientific Computing. 2002;23(6): 2041–2054. DOI: 10.1137/S1064827500379549.
- Jeltsch R, Nevanlinna O. Stability of explicit time discretizations for solving initial value problems. Numerische Mathematik. 1981;37(1):61–91. DOI: 10.1007/BF01396187.
- Jeltsch R, Nevanlinna O. Stability and accuracy of time discretizations for initial value problems. Numerische Mathematik. 1982;40(2):245–296. DOI: 10.1007/BF01400542.
- Daubechies I. Ten lectures on wavelets. Philadelphia: Society for Industrial and Applied Mathematics; 1992. 369 p. (CBMS-NSF regional conference series in applied mathematics).
- Hairer E, Nørsett SP, Wanner G. Solving ordinary differential equations I: nonstiff problems. 2nd edition. Berlin: Springer; 1993. 528 p. (Springer series in computational mathematics; volume 8). DOI: 10.1007/978-3-540-78862-1.
- Xu Y, Zhao JJ. Estimation of longest stability interval for a kind of explicit linear multistep methods. Discrete Dynamics in Nature and Society. 2010;2010:1–18. DOI: 10.1155/2010/912691.
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